Galois groups of prime degree polynomials with nonreal roots

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In the process of computing the Galois group of a prime degree polynomial f(x) over Q we suggest a preliminary checking for the existence of non-real roots. If f(x) has non-real roots, then combining a 1871 result of Jordan and the classification of transitive groups of prime degree which follows from CFSG we get that the Galois group of f(x) contains Aₚ or is one of a short list. Let f(x)∈ Q [x] be an irreducible polynomial of prime degree p ≥ 5 and r=2s be the number of non-real roots of f(x). We show that if s satisfies s (s s + 2 s + 3) ≤ p then Gal (f)= Aₚ, Sₚ.